Partial Differential Equations in Action: From Modelling to Theory

I take this opportunity to say that I have not seen any other treatise as complete as this one for a first course in partial differential equations (PDEs). It is also a perfectly suitable textbook for upper level undergraduate and first year graduate students with interests in applied mathematics, physics, and engineering. As the title suggests, the book grows from modeling fundamental processes in applied science to a theoretical study of partial differential equations. It also provides a solid theoretical foundation for numerical methods, such as finite elements.

The first part of the book comprises five chapters and covers most fundamental PDEs with solvability techniques, along with applications to physics, engineering and finance. The heat equation, the Laplace equation and wave equation and their variants are all laid out extensively. Some elegantly presented important topics, which many other books of this kind seem to either miss or briefly touch on, are uniqueness results and the maximum principle for diffusion equations, symmetric random walks, multidimensional random walks with drift and reaction, global Cauchy theory, the Black-Scholes equation, and hedging and self-financing strategy. The Laplace equation is discussed so extensively that the book covers not only well-posed problems but also a widely used method of Peron using sub/superharmonic functions. Uniqueness of solutions of the Laplace equation in unbounded exterior domains is also included. A whole chapter on scalar conservation laws and first order equations is devoted to the study of the linear transport equation, traffic dynamics, weak solutions, entropy conditions, the Riemann problem, and the method of characteristics for quasilinear equations. The chapter on waves and vibration extensively covers the wave equation along with some classical models, such as the elastic membrane and small amplitude sound waves. An application to thermoacoustic tomography is included. A detailed treatment of linear water waves is provided.

In the second part of the book, the presentation of materials undergoes abstraction and covers modern techniques for analyzing and solving PDEs. Functional analytic tools are elegantly presented, and the need of weak solutions is clearly justified. Distributions and Sobolev spaces are included as needed. Variational methods for second order elliptic problems are developed. Evolution problems and steady state equations are extensively covered as well as the application of modern techniques, such as weak formulation and energy estimates. Uniqueness and stability of abstract parabolic problems are presented as are the Faedo-Galerkin approximations. Existence, uniqueness and stability of the wave equation are also included. A chapter on systems of conservation laws includes linear hyperbolic systems, quasilinear conversation laws, and the Riemann problem for \(p\)-systems.

Some parts of the book utilize Fourier series techniques or Lebesgue measure and integration, both of which are outlined in Appendices.

Another attractive feature of the book is its richness in exercises of various levels of difficulty. At the end of each chapter, an extensive list of problems is provided. Some problems are direct applications of the theory or methods developed in the text, and other problems are to complement the arguments in proofs. There are also problems that demand a deeper understanding of the theory and methods. Steps or hints are provided for such problems. The companion book Partial Differential Equations in Action Complements and Exercises contains solutions of all problems. The text is equipped with all necessary ingredients to be used as a textbook, and will be appreciated by anyone wishing for a very complete first course in partial differential equations from modeling to theory.

Dhruba Adhikari is an Assistant Professor of Mathematics at Kennesaw State University, Georgia, USA